"A broken clock is right twice per day," is what we have been taught (assuming, as was the practice in the era when this old saw came new, that it is not a 24-hour clock forever digitally flashing something like 16:53:24). Actually, it depends on the breakage. A clock with a faulty spring which automatically jumps the minute hand forward by precisely ten minutes immediately after being set will never be right, as it will always be ten minutes ahead. A clock broken only so much that it "loses" a minute every hour will only be right once every 720 days, as it will take that long for the lost minutes to add up to a full twelve hours.

Okay, but what about a truly "broken" clock, one which sits there with hands immobilized, gears locked in stillness? What are the chances of glancing at that clock and getting the right time? If we assume an hour, minute, and second hand, with no further subdivision being paid any attention, then the chance of a person looking at any random time, and that matching to the second with the time shown, is one in 43,200. But at least those odds are substantially better than the odds of winning a lottery jackpot.

And if, by the way, the clock-glancer is not so picky, and only cares about the minute and not the second, the odds of that being right are one in 720 -- the same as the odds of looking at the clock which loses one minute per hour on a day when one may see a time which falls within a minute of being the right one. Odd, isn't it, how a totally broken clock has a better chance of being right than one only slightly broken.

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